Shapley Effects for Sensitivity Analysis With Correlated Inputs: Comparisons With Sobol' Indices, Numerical Estimation and Applications

Abstract

The global sensitivity analysis of a numerical model aims to quantify, by means of sensitivity indices estimates, the contributions of each uncertain input variable to the model output uncertainty. The so-called Sobol' indices, which are based on functional variance analysis, present a difficult interpretation in the presence of statistical dependence between inputs. The Shapley effects were recently introduced to overcome this problem as they allocate the mutual contribution (due to correlation and interaction) of a group of inputs to each individual input within the group. In this paper, using several new analytical results, we study the effects of linear correlation between some Gaussian input variables on Shapley effects, and compare these effects to classical first-order and total Sobol' indices. This illustrates the interest, in terms of sensitivity analysis setting and interpretation, of the Shapley effects in the case of dependent inputs. For the practical issue of computationally demanding computer models, we show that the substitution of the original model by a metamodel (here, kriging) makes it possible to estimate these indices with precision at a reasonable computational cost.

Shapley effects for sensitivity analysis with dependent inputs: bootstrap and kriging-based algorithms

“…• The first one is to use the CLT like Iooss and Prieur (2017). Indeed, in Castro et al (2009) the CLT gives us:…”

“…The choice of N v is independent from these values and Iooss and Prieur (2017) have also illustrated the convergence of two numerical algorithms for estimating Shapley effects.…”

“…Song et al (2016) proposed an algorithm to estimate these indices. Some studies also highlighted the potential of this kind of index in the case of correlated input, as Owen and Prieur (2017); Iooss and Prieur (2017). In this last case the Shapley effects can be a good alternative to the existing extensions of Sobol' indices mentioned above.…”

Causality and Bayesian Network PDEs for multiscale representations of porous media

“…to the assumption of independent priors in (14)). Once a range of hyperparameters is fixed, one can sample from the PDFs (16)- (18) and hence the correlation structure of P (Θ) can be computed empirically. In Figure 6, we compare the empirical correlation structure obtained from (19) over both a narrow hyperparameter range in Table 1 and a physical hyperparameter range in Table 2 to the correlation structure obtained from (14).…”

“…This framework for incorporating causal relationships through structured priors demands GSA tools that differ from the standard variance-based GSA methods. These typically assume unstructured (i.e., mutually independent) priors and are neither easy to interpret nor cheap to compute for correlated inputs [30,18]. In our application, causal relationships exist not just between parameters but also between scales.…”